Franco has a hamate injury. He left Saturday's 5-4 loss at Cincinnati after fouling off a 100.2 mph Hunter Greene pitch before striking out.
Manager Kevin Cash said Monday that the 21-year-old Franco will be sidelined for five to eight weeks. Franco is hitting .260 with five homers and 23 RBI in 58 games in the first season of a $182 million, 11-year contract.
The Rays placed Franco and center fielder Kevin Kiermaier on the 10-day injured list on Sunday.
Kiermaier, 32, is dealing with left hip inflammation. He will see a hip specialist in Tennessee after the All-Star break.
"KK's going to be down for some time," Cash said. "We need to let the hip calm down and let the expert give his recommendation. KK might have a decision to make at that point."
Kiermaier is batting .228 with seven homers and 22 RBI in 63 games this season.